R.S. Aggarwal solutions for Mathematics [English] Class 10 chapter 20 - Additional Questions [Latest edition]

Assertion (A): \(\sqrt{10}\) is an irrational number.

Reason (R): If m is a natural number which is not a perfect square then \(\sqrt{m}\) is irrational.

Assertion (A): \[\frac{19}{500}\] is a terminating decimal.

Reason (R): The rational number \[\frac{p}{q}\] is a terminating decimal, if \[q = (2^{m} \times 5^{n})\] for some whole numbers m and n.

Assertion (A): \[\frac{31}{90}\] is a nonterminating, repeating decimal.

Reason (R): The rational number \[\frac{p}{q}\] is a repeating decimal, if q has at least one factor other than 2 and 5.

Assertion (A): \[\sqrt{7}\] is an irrational number.

Reason (R): If p is prime then \[\sqrt{p}\] is irrational.

Assertion (A): \[\frac{22}{7}\] is a nonterminating, repeating decimal.

Reason (R): The rational number \[\frac{p}{q}\] is a terminating decimal, if \[q = (2^{m} \times 5^{n})\] for some whole numbers m and n.

Assertion (A): \[\frac{1}{\sqrt{3}}\] is a rational number.

Reason (R): If p is prime then \[\frac{1}{\sqrt{p}}\] is irrational.

Assertion (A): π is irrational and \[\frac{22}{7}\] is rational.

Reason (R): \[\pi \neq \frac{22}{7}\]

Assertion (A): \[\sqrt{5}\] is an irrational number.

Reason (R): Let p be a prime number and a be a positive integer. Then, p divides \[a^{2} \Rightarrow\] p divides a.

Assertion (A): \[0.\overline{6}\] is a terminating decimal.

Reason (R): The rational number \[\frac{p}{q}\] is a terminating decimal, if \[q = (2^{m} \times 5^{n})\] for some whole numbers m and n.

Assertion (A): The HCF of two numbers is 25 and their LCM is 60.

Reason (R): The HCF of two given numbers always divides their LCM exactly.

Assertion (A): The HCF of two numbers is 7 and their LCM is 70. If one of the numbers is 14 then the other number is 30.

Reason (R): For any two given positive integers a and b, we have \[\text{HCF},(a, b) \times \text{LCM},(a, b) = (a \times b).\]

Assertion (A): The number \[5^{n}\] cannot end with digit 0, where n is a natural number.

Reason (R): Prime factorisation of 5 has only two prime factors 1 and 5.

Based on the above information, answer the following questions.

1. How many books are arranged in each stack? 
2. How many stacks are used to arrange all the mathematics books? 
3. Determine the total number of stacks that will be used for arranging all the books. 
4. If the thickness of each book of history, science and mathematics is 1.8 cm, 2.2 cm and 2.5 cm respectively then find the height of each stack of history, science and mathematics books

Based on the above information, answer the following questions.

1. At what time after 8.00 a.m. will all the signals change simultaneously? 
2. At what time after 8.00 a.m. will the signals A and B change simultaneously? 
3. At what time after 8.00 a.m. will the signals B and C change simultaneously?
4. At what time after 8.00 a.m. will the signals A and C change simultaneously?

Assertion (A): A monic quadratic polynomial having 4 and –2 as its zeros is $$\left(x^{2}-2x-8\right)$$.

Reason (R): The monic quadratic polynomial having $$\alpha$$ and $$\beta$$ as its zeros, is given by $$x^{2}-(\alpha+\beta)x+\alpha\beta$$.

Assertion (A): The quadratic polynomial whose zeros are $$(3+\sqrt{2})$$ and $$(3-\sqrt{2})$$ is given by $$p(x)=(x^{2}-6x+7)$$.

Reason (R): If $$\alpha$$ is a zero of the quadratic polynomial $$p(x)$$ then $$(x-\alpha)$$ is a factor of $$p(x)$$.

Assertion (A): The polynomial $$p(x)=(x^{3}+x)$$ has one real zero.

Reason (R): A polynomial of nth degree has at most n zeros.

Assertion (A): If $$\alpha$$ and $$\beta$$ be the zeros of polynomial $$(x^{2}-6x+k)$$ such that $$(\alpha-\beta)=2$$ then $$k=8$$.

Reason (R): If $$\alpha$$ is a zero of the polynomial $$p(x)$$ then $$p(-\alpha)=0$$.

Assertion (A): If the zeros of the quadratic polynomial $$ax^{2}+bx+c$$ are both negative then a, b, c will have the same sign.

Reason (R): The zeros of a quadratic polynomial are either both positive or both negative.

Assertion (A): If one zero of the polynomial $$p(x)=(k^{2}+4)x^{2}+9x+4k$$ is the reciprocal of the other zero then $$k=2$$.

Reason (R): If $$(x-\alpha)$$ is a factor of the polynomial $$p(x)$$ then $$\alpha$$ is a zero of $$p(x)$$.

Assertion (A): If the sum of the zeros of the quadratic polynomial $$p(x)=kx^{2}+2x+3k$$, is equal to the product of its zeros then $$k=\frac{-2}{3}$$.

Reason (R): If $$\alpha$$ and $$\beta$$ be the zeros of a quadratic polynomial $$p(x)=ax^{2}+bx+c$$ then $$(\alpha+\beta)=\frac{-b}{a}$$ and $$\alpha\beta=\frac{c}{a}$$.

Assertion (A): If on dividing the polynomial $$p(x)=x^{2}-3ax+3a-7$$ by $$(x+1)$$, we get 6 as remainder then $$a=4$$.

Reason (R): When a polynomial $$p(x)$$ is divided by $$(x-\alpha)$$ then the remainder is $$p(\alpha)$$.

Assertion (A): A monic cubic polynomial having 3, 2, –1 as its zeros is $$p(x)=x^{3}+4x^{2}-x+6$$.

Reason (R): The monic cubic polynomial having $$\alpha,\beta,\gamma$$ as its zeros is given by $$x^{3}-(\alpha+\beta+\gamma)x^{2}+(\alpha\beta+\beta\gamma+\gamma\alpha)x-(\alpha\beta\gamma)$$.

Based on the above information, answer the following questions.

1. The graph of a rainbow y = f(x) is shown in the figure given below. Write the number of zeros of the curve.
   
2. If the graph of a rainbow does not intersect the x-axis but intersects y-axis at one point, then how many zeros will it have? 
3. The polynomial x² – 2x – (7a + 3) represents a rainbow. If –4 is a zero of it, find the value of a. 
4. If α and β are the zeros of the quadratic polynomialp (x) = x² – rx – s then find the value of α² + β².

Based on the above information, answer the following questions.

1. How many zeros are there for the polynomial represented by the given graph?
2. Write the zeros of the polynomial. 
3. If the zeros of a polynomial x² + (p + 1)x + q are 2 and –3 then determine the values of p and g. 
4. If the square of the difference between the zeros of the polynomial x² + ax + 45 is 144 then find the value of a.

Based on the given information, answer the following questions.

1. Find the zeros of the polynomial p(t) = 20t – 16t².
2. Which of the following types of graphs represents p(t)?
   
3. What would be the value of h at `t = 3/2`? Interpret the result.
4. How much distance has the dolphin covered before hitting the water level again?

Assertion (A): The system of equations \[2x+3y-6=0\] and \[4x+6y-12=0\] has infinitely many solutions.

Reason (R): The system of equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] has infinitely many solutions, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}.\]

Assertion (A): The system of equations \[2x-3y-5=0\] and \[4x-6y+3=0\] has no solutions.

Reason (R): The system of equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] has no solutions, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\ne\frac{c_{1}}{c_{2}}.\]

Assertion (A): The system of equations \[x+2y+2=0\] and \[3x+2y-2=0\] has a unique solution.

Reason (R): The system of equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] has a unique solution, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\ne\frac{c_{1}}{c_{2}}.\]

Assertion (A): The system of equations \[3x-6y=0\] and \[x-2y-6=0\] has infinitely many solutions.

Reason (R): The system of equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] has infinitely many solutions, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}.\]

Assertion (A): The system of equations \[2x-5y+4=0\], \[2x+y-8=0\] has a unique solution.

Reason (R): The system of equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] is inconsistent, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\ne\frac{c_{1}}{c_{2}}.\]

Assertion (A): The homogeneous system of equations \[2x+3y=0\], \[4x+6y=0\] has an infinite number of solution.

Reason (R): The homogeneous system of equations \[a_{1}x+b_{1}y=0,\ a_{2}x+b_{2}y=0\] has an infinitely many solutions, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}.\]

Assertion (A): The homogeneous system of equations \[3x-4y=0\], \[2x-3y=0\] has a unique solution x = 0 and y = 0.

Reason (R): The homogeneous system of equations \[a_{1}x+b_{1}y=0,\ a_{2}x+b_{2}y=0\] has a unique solution x = 0 and y = 0 when \[\frac{a_{1}}{a_{2}}\ne\frac{b_{1}}{b_{2}}.\]

Assertion (A): The system of linear equations \[3x+5y-4=0\], \[15x+25y-25=0\] is inconsistent.

Reason (R): The pair of linear equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] is inconsistent, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\ne\frac{c_{1}}{c_{2}}.\]

Assertion (A): The system of linear equations \[5x-9y+6=0\], \[10x-18y+12=0\] is inconsistent.

Reason (R): The pair of linear equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] is inconsistent, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\ne\frac{c_{1}}{c_{2}}.\]

Assertion (A): The system of linear equations \[2x-3y+1=0\], \[4x-6y+3=0\] is inconsistent.

Reason (R): The system of linear equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] has a unique solution, if \[\frac{a_{1}}{a_{2}}\ne\frac{b_{1}}{b_{2}}.\]

Assertion (A): The system of linear equations \[x-2y+6=0\], \[2x-4y+9=0\] has a unique solution.

Reason (R): The system of linear equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] has a unique solution, if \[\frac{a_{1}}{a_{2}}\ne\frac{b_{1}}{b_{2}}.\]

Assertion (A): The system of linear equations \[x+y-9=0\], \[x-y+3=0\] has a unique solution.

Reason (R): The system of linear equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] has a unique solution, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}.\]

Based on the above information, answer the following questions.

(i) What are the fixed charges?

(ii) What are the charges per kilometre?

(iii) If fixed charge are ₹20 and charges per kilometre are ₹10 then how much would Arun have to pay for travelling a distance of 10 km?

(iv) Find the total amount paid by Arun for travelling 10 km from home to office and 25 km from office to home. [Fixed charges and charges per kilometre are as in (i) and (ii).]

Based on the above information, answer the following questions.

1. Represent the above information algebraically (in terms of p and q). 
2. What is the prize amount for hockey? 
3. Prize amount for which game is higher and by how much? 
4. What will be the total prize amount if there are 2 students each from the two games?

Based on the above information, answer the following questions.

1. Represent the given information in terms of x and y. 
2. Find the monthly fee paid by a poor child. 
3. Find the difference in the monthly fee paid by a poor child and a rich child. 
4. If there are 10 poor and 20 rich children in batch II, then what is the monthly collection of fees from batch II?

Assertion (A): The quadratic equation $$x^{2}+3x+4=0$$ has both real roots.

Reason (R): The equation $$ax^{2}+bx+c=0,\ a\neq 0$$ has both real roots, if $$(b^{2}-4ac)\geq 0$$.

Assertion (A): The polynomial $$p(x)=x^{2}+3x+3$$ has two real zeros.

Reason (R): A quadratic polynomial can have at most two real zeros.

Assertion (A): The polynomial $$x^{2}+4x$$ has two real zeros.

Reason (R): The zeros of the polynomial $$x^{2}+ax\ (a\neq 0)$$ are 0 and $$a$$. 

Assertion (A): If $$(5 + \sqrt{7})$$ is a root of a quadratic equation with rational coefficients then the other root is $$(5 - \sqrt{7})$$.

Reason (R): The surd roots of a quadratic equation with rational coefficients occur in conjugate pairs.

Assertion (A): If one root of the quadratic equation $$4x^{2}-10x+(k-4)=0$$ is the reciprocal of the other then the value of $$k$$ is 8.

Reason (R): The roots of the quadratic equation $$x^{2}-x+1=0$$ are real.

Assertion (A): If the equation $$x^{2}-kx+1=0$$ has no real roots then $$-2<k<2$$.

Reason (R): The equation $$ax^{2}+bx+c=0$$ has real roots, if $$(b^{2}-4ac)\geq 0$$.

Assertion (A): The quadratic equation $$\sqrt{3}x^{2}+11x+6\sqrt{3}=0$$ has both real roots.

Reason (R): The quadratic equation $$ax^{2}+bx+c=0$$ has both imaginary roots, if $$(b^{2}-4ac)<0$$.

Assertion (A): If –5 is a root of $$2x^{2}+2px-15=0$$ and the equation $$p(x^{2}+x)+k=0$$ has equal roots then $$k=\frac{7}{8}$$.

Reason (R): The equation $$ax^{2}+bx+c=0,\ a\neq 0$$ has equal roots, if $$(b^{2}-4ac)=0$$.

Assertion (A): If the quadratic equation $$px^{2}-4x+4=0$$ has real roots them $$p\leq 1$$.

Reason (R): The quadratic equation $$ax^{2}+bx+c=0$$ has both real roots, if $$(b^{2}-4ac)<0$$.

Assertion (A): The quadratic equation $$x^{2}+5kx+16=0$$ has no real roots, if $$\frac{-8}{5}<k<\frac{8}{5}$$.

Reason (R): The equation $$ax^{2}+bx+c=0,\ a\neq 0$$ has no real root, if $$(b^{2}-4ac)<0$$.

Assertion (A): The quadratic equation $$x^{2}-x-6=0$$ has –2 and 3 as its roots.

Reason (R): If $$ax^{2}+bx+c=0$$ then $$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.$$

Assertion (A): The quadratic equation $$x^{2}-4x+3=0$$ has both real roots.

Reason (R): The equation $$ax^{2}+bx+c=0,\ a\neq 0$$ has both real roots, if $$(b^{2}-4ac)\geq 0$$.

Assertion (A): The quadratic equation $$3x^{2}-5x+4=0$$ has no real root.

Reason (R): The equation $$ax^{2}+bx+c=0,\ a\neq 0$$ has no real root, if $$(b^{2}-4ac)<0$$.

Based on the given information, answer the following questions.

1. Taking x as the number of rows in the beginning, represent the above situation by a quadratic equation. 
2. How many rows are there in the original arrangement? 
3. How many seats are there in the auditorium in the beginning? 
4. How many seats are there in the auditorium after rearrangement?

 Based on the above information, answer the following questions.

1. Write an algebraic equation depicting the above information. 
2. Write the corresponding quadratic equation in standard form. 
3. What should be the new dimensions of the enlarged photo? 
4. Can any rational value of x make the new area equal to 220 cm²?

Based on the given information, answer the following questions.

1. Write a quadratic equation in x to represent the above information. 
2. Find the value of x.
3. Find the perimeter of rectangle PQRS. 
4. For what value of x will the area of the path of uniform width be 93600 m²?

Assertion (A): The numbers $$a, b, c$$ are in AP if and only if $$2b = (a + c).$$

Reason (R): The sum of the first $$n$$ odd natural numbers is $$n^{2}.$$

Assertion (A): The sum of first n natural numbers is given by $$S_{n} = \frac{1}{2}n(n + 1).$$

Reason (R): In an AP with first term a and common difference d, the sum of n terms is given by $$S_{n} = \frac{n}{2} \times [2a + (n - 1)d].$$

Assertion (A): The sum of first n even natural numbers is n(n + 1).

Reason (R): The sum of n terms of an AP with first term a and last term $$l$$ is $$S_{n} = \frac{n}{2},(a + l).$$

Assertion (A): The 9th term of the AP 37, 33, 29, 25, ... is –5.

Reason (R): The nth term of the AP with first term a and common difference d is given by $$T_{n} = a + (n - 1)d.$$

Assertion (A): The 10th term from the end of the AP 16, 11, 6, 1, ..., –79 is –34.

Reason (R): The nth term from the end of an AP with last term $$l$$ and common difference d is $${\{l - (n - 1)d}\}.$$

Assertion (A): 47 + 45 + 43 + ... + 11 is an AP having 19 terms.

Reason (R): In an AP with first term a and common difference d, the nth term is given by $$T_{n} = a + (n - 1)d.$$

Assertion (A): In an AP, the first term is 6, the last term is 78 and the sum of the given terms is 630. Then, there are 16 terms in the given AP.

Reason (R): If a is the first term, $$l$$ is the last term and n is the number of terms in an AP then $$S_{n} = \frac{n}{2} (a + l).$$

Assertion (A): The next term of the AP $$\sqrt{3},\ \sqrt{12},\ \sqrt{27},\ \ldots$$ is $$\sqrt{48}.$$

Reason (R): In an AP, we have $$T_{n + 1} = (T_{n} + d).$$

Assertion (A): In an AP, it is given that $$T_{9} = 39$$ and $$T_{15} = 63.$$ Then, its first term is 7 and the common difference is 4.

Reason (R): In an AP with first term a and common difference d, the nth term is given by $$T_{n} = a + (n - 1)d.$$

Assertion (A): The sum of 14 terms of the AP 7 + 10 + 13 + ... is 375.

Reason (R): The sum of n terms of an AP with first term a and common difference d is given by $$S_{n} = \frac{n}{2} \cdot {\{2a + (n - 1)d}\}.$$

Based on the given information, answer the following questions.

1. Find the number of pots placed in the 10th row.
2. Find the difference in the number of pots placed in the 5th row and the 2nd row.
3. If Sanjana wants to place 100 pots in total then find the total number of rows formed in the arrangement.
4. If Sanjana has sufficient space for 12 rows then how many total number of pots are placed by her with the same arrangement?

Based on the above information, answer the following questions.

1. In which year will the factory produce 29200 sets?
2. Find the production during the 8th year.
3. Find the production during the first 3 years.
4. Find the difference in production during the 7th year and the 4th year.

Based on the above information, answer the following questions.

1. If the first circular row has 30 seats then how many seats will there be in the 10th row?
2. For 1500 seats, how many rows must the auditorium have?
3. If 1500 seats are to be arranged in the auditorium, how many seats are still left to be put after the 10th row?
4. If there were 17 rows in the auditorium, how many seats would be in the middle row?

Based on the given information, answer the following questions.

1. Find the amount paid by him in the 10th instalment.
2. Find the total amount paid by him in the first 10 instalments.
3. In how many instalments would he clear his total loan?
4. What amount will he be able to clear in his first 45 instalments?

Based on the given figure, answer the following questions.

1. Identify the AP for the number of squares in each row.
2. Identify the AP for the number of triangles in each row.
3. If each shaded square is of side 2 cm then find the shaded area when 15 rows have been designed.
4. Write a formula for finding the total number of triangles in n number of rows. Hence, find \(S_{10}\).

1. Write the AP for the number of triangles used in the figures. Also, write the (n)th term of this AP.
2. Write the AP for the number of matchsticks used in the figures. Also, write the (n)th term of this AP.
3. Find the total number of triangles used in the first 20 figures.
4. Which figure has 61 matchsticks?

Based on the above information, answer the following questions.

1. How many throws did Sanjitha practice on the 11th day of the camp?
2. What would be Sanjitha’s throw distance at the end of 6 weeks?
3. When will she be able to achieve a throw of 11.16 m?
4. How many throws did she make during the entire camp of 15 days?

Assertion (A): The point $$P(0, 2)$$ is the point of intersection of the $$y$$-axis with the line $$3x + 2y = 4$$.

Reason (R): The distance of the point $$P(0,2)$$ from the $$x$$-axis is 2 units.

Assertion (A): If the points $$A(4,3)$$ and $$B(x, 5)$$ lie on a circle with centre $$O(2, 3)$$ then the value of $$x$$ is 2.

Reason (R): The centre of a circle is the midpoint of each chord of the circle.

Assertion (A): A point on the $$x$$-axis is of the form $$(x, 0)$$ and a point on the $$y$$-axis is of the form $$(0, y)$$.

Reason (R): The abscissa of a point denotes its distance from $$x$$-axis and the ordinate denotes its distance from $$y$$-axis.

Assertion (A): The points $$A(-1, 1), B(5, 7)$$ and $$C(8, 10)$$ cannot be the vertices of a triangle.

Reason (R): $$\operatorname{area}(\triangle ABC) = 0$$.

Assertion (A): The points $$A(2, -2), B(-3, 8)$$ and $$C(-1, 4)$$ are collinear.

Reason (R): $$BC + AC = AB$$.

Assertion (A): The points $$A(1, 2), B(-2, 3)$$ and $$C(-3, -4)$$ are not collinear.

Reason (R): $$\operatorname{ar}(\triangle ABC) = 11$$ sq units.

Assertion (A): The points $$A(0, 3), B(0, -5)$$ and $$C(2, 3)$$ are noncollinear.

Reason (R): $$A$$ and $$B$$ lie on $$x$$-axis and $$C$$ lies in first quadrant.

Assertion (A): The points $$A(1, -2), B(2, 3), C(-3, 2)$$ and $$D(-4, -3)$$ are the vertices of a parallelogram.

Reason (R): The midpoints of both the diagonals $$AC$$ and $$BD$$ are $$(-1, 0)$$.

Assertion (A): The points $$A(0, -4), B(6, 2), C(3, 5)$$ and $$D(-3, -1)$$ taken in that order are the vertices of a rectangle $$ABCD$$.

Reason (R): The midpoint of both $$AC$$ and $$BD$$ is $$\left(\frac{3}{2}, \frac{1}{2}\right)$$.

Assertion (A): Point $$P(5, -3)$$ is one of the points of trisection of the line segment joining the points $$A(7, -2)$$ and $$B(1, -5)$$.

Reason (R): $$P$$ divides $$AB$$ in the ratio $$2 : 1$$.

Assertion (A): The centroid of $$\triangle ABC$$ having vertices $$A(-1, 0), B(5, -2)$$ and $$C(8, 2)$$ is $$G(4, 0)$$.

Reason (R): The centroid of $$\triangle ABC$$ having vertices $$A(x_{1}, y_{1}), B(x_{2}, y_{2})$$ and $$C(x_{3}, y_{3})$$ is given by $$G = \left(\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3}\right).$$

Assertion (A): The point $$P(-2, 4)$$ lies on a circle of radius 6 and centre $$C(3, 5)$$.

Reason (R): $$PC < 6$$.

Assertion (A): The point $$P(-3, 2)$$ lies on the line segment joining the points $$A(-3, 5)$$ and $$B(-3, -5)$$.

Reason (R): The three points $$A, B$$ and $$P$$ lie on the line $$x = -4$$.

Based on the above information, answer the following questions.

1. Taking O as the origin, the coordinates of P are (–200, 0) and that of Q are (200, 0). PQRS being a square, what are the coordinates of R and S?
2. What is the area of square PQRS?
3. What is the length of diagonal PR in square PQRS?
4. If S divides CA in the ratio k : 1, what is the value of k, where point (A) is (200, 800)?
5. What are the coordinates of C ?

Look at the given figure and answer the following questions.

1. What are the coordinates of the point W?
2. What are the coordinates of the point R?
3. What is the distance between the points W and R?
4. What are the coordinates of the point which is exactly halfway between the line segment joining W and R?
5. What are the coordinates of the point which is at about one fourth distance in the line segment joining W and R?

Based on the above information, answer the questions given below.

1. What are the coordinates of the point of intersection of diagonals AC and BD?
2. What is the length of diagonal AC?
3. Find the ratio AB : AD.
4. Find the area of the campaign board ABCD.
5. What is the perimeter of the campaign board ABCD?

Using the picture of a hockey field given above, answer the questions that follow.

1. What are the coordinates of the centroid of \(\triangle HFJ\)?
2. What are the coordinates of the point on the x-axis, equidistant from F and I?
3. What are the coordinates of the point on the y-axis, equidistant from B and C?
4. What are the coordinates of the position of a player Q such that his distance from K is twice his distance from F, where K, Q and F are collinear?
5. If a player P needs to be at equal distances from A and G such that A, P and G are in a straight line then what will be the position of P?

Assertion (A): If $$\triangle FED \sim \triangle STU$$ then $$\frac{DE}{ST}=\frac{EF}{TU}$$.

Reason (R): The corresponding sides of similar triangles are proportional.

Assertion (A): Two quadrilaterals are similar, if their corresponding angles are equal.

Reason (R): Two triangles are similar by AAA-criterion if their corresponding angles are equal.

Assertion (A): In the given figure, if $$DE\parallel BC$$ then $$\triangle ADE\sim\triangle ACB$$.

Reason (R): Being corresponding angles, we have $$\angle D=\angle B$$ and $$\angle E=\angle C$$.

Assertion (A): If $$\triangle ABC\sim\triangle DEF$$, $$\angle A=47^{\circ}$$ and $$\angle E=83^{\circ}$$ then $$\angle C=50^{\circ}$$.

Reason (R): Similar triangles are always equiangular.

Assertion (A): The three angles of a triangle are in the ratio 1 : 2 : 3. Then, the triangle is right angled.

Reason (R): The measures of the three angles of the triangle 30°, 60° and 90°.

Assertion (A): In the given figure, if $$\triangle ABC\sim\triangle DEF$$ then $$x = 5$$.

Reason (R): $$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$$.

Assertion (A): Two sides and the perimeter of one triangle are respectively three times the corresponding sides and the perimeter of the other triangle. Then, the two triangles are similar.

Reason (R): If all the three corresponding sides of two triangles are proportional then the triangles are similar.

Assertion (A): In $$\triangle ABC$$, $$AB=5\ \text{cm}$$, $$BC=12\ \text{cm}$$ and $$AC=13\ \text{cm}$$. Then, $$\angle B=90^{\circ}$$.

Reason (R): $$AB^{2}+AC^{2}=BC^{2}$$.

Assertion (A): In the given figure, if $$\angle AED=\angle ABC$$ then $$\triangle ADE\sim\triangle ACB$$.

Reason (R): If two corresponding angles of two triangles are equal then the triangles are similar.

Assertion (A): In the given figure, $$MN\parallel BC$$ and $$AM : MB = 1 : 2$$. Then, $$\frac{\operatorname{ar}(\triangle AMN)}{\operatorname{ar}(\triangle ABC)}=\frac{1}{4}$$.

Reason (R): The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

Assertion (A): Two isosceles triangles with equal vertical angles are similar.

Reason (R): If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional then the two triangles are similar.

Assertion (A): In $$\triangle ABC$$, $$P$$ and $$Q$$ are points on $$AB$$ and $$AC$$ respectively such that $$AP = 3\ \text{cm}$$, $$PB = 6\ \text{cm}$$, $$AQ = 5\ \text{cm}$$ and $$QC = 10\ \text{cm}$$. Then, $$BC = 3PQ$$.

Reason (R): The corresponding sides of similar triangles $$APQ$$ and $$ABC$$ are proportional.

Assertion (A): In $$\triangle ABC$$ and $$\triangle PQR$$, if $$\frac{AB}{QR} = \frac{BC}{PR} = \frac{CA}{PQ}$$ then $$\triangle ABC\sim\triangle PQR$$.

Reason (R): If the corresponding sides of two triangles are proportional then the triangles are similar.

Assertion (A): If $$\triangle ABC\sim\triangle DEF$$, $$\operatorname{ar}(\triangle ABC) = 81\ \text{cm}^{2}$$, $$\operatorname{ar}(\triangle DEF) = 144\ \text{cm}^{2}$$ and $$EF = 24\ \text{cm}$$ then $$BC = 18\ \text{cm}$$.

Reason (R): $$\frac{\operatorname{ar}(\triangle ABC)}{\operatorname{ar}(\triangle DEF)}=\frac{AB^{2}}{DE^{2}}=\frac{BC^{2}}{EF^{2}}=\frac{AC^{2}}{DF^{2}}$$.

Assertion (A): If $$\triangle ABC\sim\triangle DEF$$ such that $$2AB = 3DE$$ and $$BC = 6\ \text{cm}$$ then $$EF = 4\ \text{cm}$$.

Reason (R): The corresponding sides of similar triangles are proportional.

1. Find Abhay's height (in cm).
2. Find the length of Abhay's shadow (in metres) when he is at a distance of 12 metres from the foot of the pole.
3. When Abhay is 12 metres away from the foot of the pole, he meets his friend Sachin who then moves towards the pole at the rate of 5 cm/s. If after 2 minutes the length of Sachin's shadow is 1.2 metres, what is his height?
4. Name the similarity criterion used in solving the above problem.

Assertion (A): A tangent to a circle is perpendicular to the radius through the point of contact.

Reason (R): The lengths of tangents drawn from an external point to a circle are equal.

Assertion (A): The tangents drawn at the end points of a diameter of a circle are parallel to each other.

Reason (R): A tangent to a circle is perpendicular to the radius through the point of contact.

Assertion (A): If PA and PB are tangents drawn from an external point P to а circle with centre O then the quadrilateral AOBP is cyclic.

Reason (R): The angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.

Assertion (A): In the given figure, PQ and PR are tangents to a circle centred at О and S is any point on the circle. If ∠QPR = 55° then ∠QSR = 62.5°.

Reason (R): If a chord AB subtends an angle of 60° at the centre of a circle then the angle between the tangents at A and B is also 60°.

Assertion (A): In the given figure, PQ is a tangent to a circle centred at O. If the radius of the circle is 5 cm then the length of the tangent PQ is \[\dfrac{5}{\sqrt{3}}\ \text{cm.}\]

Reason (R): If two tangents are drawn to a circle from an external point then they subtend equal angles at the centre.

Assertion (A): In the given figure, the quadrilateral PQRS circumscribes a circle. Then, PA + CS = PS.

Reason (R): The tangents drawn at the end points of a diameter of a circle are parallel to each other.

Assertion (A): If PA and PB are tangents to the circle centred at O and ∠OPA = 30° then ΔPAB is an equilateral triangle.

Reason (R): The lengths of tangents drawn from an external point to a circle are equal.

Assertion (A): For \(0<\theta \leq 90^{\circ},(\operatorname{cosec} \theta-\cot \theta)\) and \((\operatorname{cosec} \theta+\cot \theta)\) are reciprocal of each other.

Reason (R): \((\operatorname{cosec}^{2} \theta-\cot ^{2} \theta)=1\).

Assertion (A): If \((\tan \theta+\cot \theta)=n\) then \(\tan ^{2} \theta+\cot ^{2} \theta=n^{2}\).

Reason (R): \(\cot \theta=\frac{1}{\tan \theta}\).

Assertion (A): \(\tan 48^{\circ}=\cot 52^{\circ}\).

Reason (R): \(\tan \theta=\cot \left(90^{\circ}-\theta\right)\) for \(0^{\circ}<\theta<90^{\circ}\).

Assertion (A): If \(\cos 2 \theta=\sin \left(\theta-15^{\circ}\right)\) then \(\theta=35^{\circ}\).

Reason (R): \(\cos \theta=\sin \left(90^{\circ}-\theta\right)\), for \(0^{\circ} \leq \theta \leq 90^{\circ}\).

Assertion (A): The value of \((\sin 80^{\circ}-\cos 80^{\circ})\) is positive.

Reason (R): \(45^{\circ}<\theta \leq 90^{\circ} \Rightarrow \sin \theta>\cos \theta\).

Assertion (A): The value of \((\cos ^{2} 24^{\circ}-\sin ^{2} 66^{\circ})\) is positive.

Reason (R): \(0^{\circ} \leq \theta \leq 90^{\circ} \Rightarrow \cos \left(90^{\circ}-\theta\right)=\sin \theta\).

Assertion (A): The value of \((\sin \theta+\cos \theta)\) is always greater than 1.

Reason (R): For \(0^{\circ} \leq \theta \leq 90^{\circ}\), the value of \(\sin \theta\) varies from 0 to 1 and the value of \(\cos \theta\) varies from 1 to 0.

Assertion (A): The value of \(\tan 1^{\circ} \cdot \tan 11^{\circ} \cdot \tan 21^{\circ} \cdot \tan 69^{\circ} \cdot \tan 79^{\circ} \cdot \tan 89^{\circ}\) is 1.

Reason (R): For \(0^{\circ}<\theta<90^{\circ}\), we have \(\tan \left(90^{\circ}-\theta\right)=\cot \theta\).

Assertion (A): If \((\operatorname{cosec} \theta-\cot \theta)=\frac{1}{3}\) then \((\operatorname{cosec} \theta+\cot \theta)=3\).

Reason (R): \(\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1\) for all values of \(\theta\), where \(0^{\circ} \leq \theta \leq 90^{\circ}\).

Assertion (A): If \(\operatorname{cosec} \theta=\frac{3}{2}\) then \((\operatorname{cosec}^{2} \theta+\cot ^{2} \theta)=\frac{7}{4}\).

Reason (R): \(\cot ^{2} \theta=\left(\operatorname{cosec}^{2} \theta-1\right)\), where \(0^{\circ} \leq \theta \leq 90^{\circ}\).

Assertion (A): The value of \((\cot 10^{\circ} \times \cot 30^{\circ} \times \cot 80^{\circ})\) is \(\frac{\sqrt{3}}{2}\).

Reason (R): \(\cot \left(90^{\circ}-\theta\right)=\tan \theta\).

Assertion (A): If \(\sec 4 \theta=\operatorname{cosec}\left(\theta-10^{\circ}\right)\) and \(4 \theta\) is acute then \(\theta=20^{\circ}\).

Reason (R): \(\sec \phi=\operatorname{cosec}\left(90^{\circ}-\phi\right)\) for all acute angles \(\phi\).

Assertion (A): If \(\sin A=\frac{8}{17}\) then \(\cot A=\frac{15}{8}\).

Reason (R): In a \(\triangle A B C\), right angled at B, we have \(\sin A=\frac{B C}{A C}\) and \(\cot A=\frac{A B}{B C}\). 

Assertion (A): If \(\sec \theta=\frac{25}{7}\) then \(\operatorname{cosec} \theta=\frac{25}{24}\).

Reason (R): In a \(\triangle A B C\), right angled at B, we have \(\sec \theta=\frac{A C}{A B}\) and \(\operatorname{cosec} \theta=\frac{A C}{B C}\). 

Assertion (A): If \(\sin \alpha=\frac{1}{2}\) then \((3 \cos \alpha - 4 \cos^{3} \alpha) = 0\).

Reason (R): \(\cos 3 \alpha = 3 \cos \alpha - 4 \cos^{3} \alpha\).

Based on the above information, answer the following questions.

1. Draw a neat well-labelled figure based on the above information.
2. Find the height of the poles.
3. If the distances of the top of the poles from the point of observation are taken as p and q then find a relation between p and q.
4. Find the location of the point of observation.

Radio towers are used for transmitting a range of communication services including radio and television. The tower will either act as an antenna itself or support one or more antennas on its structure. On a similar concept, a radio station tower was built in two sections A and B. The tower is supported by wires from a point O. The distance between the base of the tower and point O is 36 cm. From point O, the angle of elevation of the top of the section B is 30° and the angle of elevation of the top of section A is 45°.

Based on the given information, answer the following questions.

1. Find the length of the wire from the point O to the top of section B.
2. Find the distance AB.
3. Find the area of ΔOPB.
4. Find the height of section A from the base of the tower.

Based on the above information, answer the following questions.

1. Taking h metres as the height of each lamp post and x metres as the distance between the feet of the two lamp posts, find a relation between x and h.
2. After moving 15 m towards one lamp post, what is the relation between x and h?
3. Find the height of each lamp post.
4. Find the distance between the two lamp posts.

Based on the above information, answer the following questions.

1. Draw a neat and well-labelled figure to represent the above information.
2. What is the distance travelled by the plane in 30 seconds?
3. Keeping the height constant, during the above flight, it was observed that after \(15(\sqrt{3}-1)\) seconds, the angle of elevation changed to \(45^{\circ}\). How much is the distance travelled in that duration?
4. What is the speed of the plane in km/hr?

Based on the above information, answer the following questions.

1. At what distance from the foot of the tree was Ratan observing the bird sitting on the tree?
2. How far did the bird fly in the mentioned time?
3. After hitting the tree, how far did the ball travel in the sky when Ratan saw it?
4. What is the speed of the bird in metre/minute if it has flown \(20(\sqrt{3}+1) \mathrm{~m}\)?

Assertion (A): The area of a semicircle of diameter $$d$$ is $$\frac{1}{8}\pi d^{2}$$.

Reason (R): The central angle of a semicircle is $$90^{\circ}$$.

Assertion (A): If the circumferences of two circles are in the ratio $$5 : 6$$ then the ratio of their radii is $$5 : 6$$.

Reason (R): The circumference of a circle is directly proportional to the square of the radius of the circle.

Assertion (A): If the radius of a semicircular protractor is $$7\ \text{cm}$$ then its perimeter is $$36\ \text{cm}$$.

Reason (R): The perimeter of a semicircular protractor of radius $$r$$ cm is $$r(\pi + 2)$$ cm.

Assertion (A): If the radius of a circle is tripled then the resulting percentage increase in the area of the circle is $$900\%$$.

Reason (R): The area of a circle is directly proportional to the square of its radius.

Assertion (A): If a circle and a square have the same perimeter then the area of the square is greater than the area of the circle.

Reason (R): If the radius of the circle is $$r$$ and the side of the square is $$s$$, then $$r = \frac{2s}{\pi}$$.

Assertion (A): If a bicycle wheel makes $$5000$$ revolutions in moving $$11\ \text{km}$$ then the diameter of the wheel is $$70\ \text{cm}$$.

Reason (R): The distance covered by a wheel in $$1$$ revolution is equal to the circumference of the wheel.

Assertion (A): The area of the square inscribed in a circle of radius $$10\ \text{cm}$$ is $$200\ \text{cm}^{2}$$.

Reason (R): Diagonal of inscribed square = diameter of circle.

Assertion (A): The length of the arc of a sector of a circle with radius $$14\ \text{cm}$$ and of central angle $$60^{\circ}$$ is $$\frac{88}{3}\ \text{cm}$$.

Reason (R): The length of the arc of a circle of radius $$r$$ cm and central angle $$\theta$$ is $$\left(\frac{2\pi r\theta}{360}\right)\ \text{cm}$$.

Assertion (A): OACB is a quadrant of a circle with centre O and radius $$7\ \text{cm}$$, where ACB is the arc. Then, the perimeter of the quadrant is $$25\ \text{cm}$$.

Reason (R): The perimeter of the sector of a circle of radius $$r$$ cm and central angle $$\theta$$ is $$2r\left(1 + \frac{\pi\theta}{360}\right)\ \text{cm}$$.

Assertion (A): The area of the sector of a circle of radius $$14\ \text{cm}$$ and central angle $$90^{\circ}$$ is $$154\ \text{cm}^{2}$$.

Reason (R): The area of the sector of a circle of radius $$r$$ cm and central angle $$\theta$$ is $$\left(\pi r^{2}\frac{\theta}{360}\right)\ \text{cm}^{2}$$.

Assertion (A): If a sector of a circle with central angle $$60^{\circ}$$ has an area $$924\ \text{cm}^{2}$$ then the radius of the circle is $$42\ \text{cm}$$.

Reason (R): The sum of the areas of a major sector of a circle and its corresponding minor sector is equal to the area of the circle.

Assertion (A): The area of the sector of a circle of radius $$21\ \text{cm}$$ cut-off by an arc of length $$22\ \text{cm}$$ is $$231\ \text{cm}^{2}$$.

Reason (R): The angle subtended by the arc at the centre of the circle is $$30^{\circ}$$.

Assertion (A): The hour hand of a clock is $$6\ \text{cm}$$ long. The angle swept by it between 9.20 a.m. and 9.55 a.m. is $$35^{\circ}$$.

Reason (R): To find the area swept by the hour hand of the clock in a certain time interval, we calculate the area of the sector of a circle.

Assertion (A): The difference between the areas of two segments of a circle of radius $$5\ \text{cm}$$, formed by a chord subtending an angle of $$90^{\circ}$$ at the centre, is $$\left(\frac{25}{2}\pi + 25\right)\ \text{cm}^{2}$$.

Reason (R): The area of a segment of a circle is less than the area of the corresponding sector.

Based on the above information, answer the following questions.

1. Find the area of the square ABCD.
2. What is the area of the circle drawn inside the square?
3. If the circle and the four quadrants are cut off from the square ABCD and removed then find the area of the remaining portion of the square ABCD.
4. Find the combined area of the four quadrants and the circle removed.

Based on the given information, answer the following questions.

1. What is the total perimeter of the parking area?
2. What is the total area of the parking and the two quadrants?
3. What is the ratio of the area of the playground to the area of the parking area?
4. Find the cost of fencing the playground and parking area at the rate of ₹ 2 per unit.

Based on the above information, answer the following questions.

1. What is the area of the quadrant ODCO?
2. Find the area of the \(\triangle AOB\).
3. What is the total cost of silver plating the shaded part ABCD?
4. What is the length of \(\operatorname{arc} CD\)?

1. length of the chord AB.
2. area of the \(\triangle OAB\).
3. area of the segment ACBA.
4. perimeter of the sector OACBO.

Based on the above information, answer the following questions.

1. Write an equation representing the total area of the two segments in terms of r.
2. Find the value of r.
3. Find the area of the segment with red roses.
4. Find the area of the segment with yellow flowers.

Assertion (A): The diameter of a sphere, whose surface area is $$616\ \mathrm{cm}^{2}$$, is 7 cm.

Reason (R): The surface area of a sphere of radius $$r$$ is $$4\pi r^{2}$$.

Assertion (A): The curved surface area of a cone having height 24 cm and radius 7 cm is $$550\ \mathrm{cm}^{2}$$.

Reason (R): The curved surface area of a cone is given by $$S = 2\pi rh$$, where $$r$$ and $$h$$ denote the radius of the base and height of the cone respectively.

Assertion (A): The total surface area of a solid hemisphere of diameter $$d$$ is $$\frac{3}{4}\pi d^{2}$$.

Reason (R): If the diameter of a solid hemisphere is halved then its total surface area reduces to 25% of the original value.

Assertion (A): If the curved surface area of a right circular cylinder of height 14 cm is $$88\ \mathrm{cm}^{2}$$ then the diameter of its circular base is 2 cm.

Reason (R): Curved surface area of a right circular cylinder = area of circular base × height.

Assertion (A): The volume of a right circular cone whose area of the base is $$156\ \mathrm{cm}^{2}$$ and the vertical height is 8 cm, is $$416\ \mathrm{cm}^{3}$$.

Reason (R): If the radius and height of a cone are doubled then its volume becomes four times.

Assertion (A): If the volumes of two spheres are in the ratio 125 : 64 then the ratio of their surface areas is 25 : 16.

Reason (R): If the volume and surface area of a sphere are numerically equal then the radius of the sphere is 3 units.

Assertion (A): The surface area of the largest sphere that can be inscribed in a hollow cube of side $$a\ \mathrm{cm}$$ is $$\pi a^{2}\ \mathrm{cm}^{2}$$.

Reason (R): The surface area of a sphere of radius $$r$$ is $$\frac{4}{3}\pi r^{3}$$.

Assertion (A): A solid is in the form of a cone of radius $$r$$ surmounted on a hemisphere of the same radius. If the height of the cone is the same as the diameter of its base then the volume of the solid is $$\frac{4}{3}\pi r^{3}$$.

Reason (R): Total volume of the solid $$=\text{volume of the cone}+\text{volume of the hemisphere}$$

Assertion (A): The surface area of the cuboid formed by joining two cubes of sides 4 cm each, end to end, is $$160\ \mathrm{cm}^{2}$$.

Reason (R): The surface area of a cuboid of dimensions $$l \times b \times h$$ is $$(lb + bh + lh)$$.

Assertion (A): The volume of the largest sphere carved out of a solid cube of side 21 cm is $$4851\ \mathrm{cm}^{3}$$.

Reason (R): The radius of the largest sphere carved out of a solid cube is half the side of the cube.

Assertion (A): A top is of the shape of a cone mounted over a hemisphere of the same radius. If the radius of the hemisphere is 3.5 cm and the total height of the top is 15.5 cm then the total surface area of the top is $$214.5\ \mathrm{cm}^{2}$$.

Reason (R): The total surface area of the top is the sum of the curved surface area of the hemisphere and the curved surface area of the cone.

Assertion (A): A hemispherical bowl is made of steel of thickness 1 cm. If the inner radius of the bowl is 5 cm then the volume of steel used is $$\left(\frac{182}{3}\pi\right)\ \mathrm{cm}^{3}$$.

Reason (R): The volume of steel is calculated as the difference between the volumes of two hemispheres with radii 6 cm and 5 cm.

Assertion (A): If a conical cup has the same radius but half the height as that of a cylindrical cup then the ratio of the volume of the conical cup to that of the cylindrical cup is 2 : 3.

Reason (R): The volume of a cone is one third of the volume of a cylinder of the same base radius and same height.

Assertion (A): The total surface area (internal and external) of a hemispherical bowl having radii $$r_{1}$$ and $$r_{2}\ (r_{2} > r_{1})$$ is $$3\pi r_{1}^{2} + \pi r_{2}^{2}$$.

Reason (R): The total surface area of a solid hemisphere of radius $$r$$ is $$3\pi r^{2}$$.

Assertion (A): 168 spherical shots of diameter 3 cm each can be made by melting a cuboidal solid of dimensions $$18\ \mathrm{cm} \times 22\ \mathrm{cm} \times 6\ \mathrm{cm}$$.

Reason (R): Total volume of the spherical shots = volume of the cuboidal solid.

Based on the above information, answer the following questions.

1. Find the area of the base of the cylindrical cup.
2. What is the capacity of the hemispherical cup?
3. What is the capacity of the cylindrical cup?
4. Find the curved surface area of the cylindrical cup.

Based on the above information, answer the following questions. [Use π = 3.14]

1. What is the volume of the material used for making the mallet?
2. The bowl is to be polished from inside. Find the inner surface area of the bowl.
3. Find the volume of metal used to make the bowl.
4. Find the total surface area of the mallet.

Based on the above information, answer the following questions.

1. Find the volume of 50 toys.
2. Find the ratio of the volume of hemisphere to the volume of cone.
3. Find the cost of painting the hemispherical part of the 50 toys at the rate of ₹10 per cm².
4. If one such toy is melted and recast into a solid cylinder of diameter 28 cm then find the height of the cylinder.

Based on the above information, answer the following questions.

1. Find the volume of the wood carved out to make one cylindrical hollow.
2. Find the lateral surface area of the cuboid to paint it with green colour.
3. Find the volume of wood in the remaining cuboid after carving out the seven cylindrical hollows.
4. Find the surface area of the top surface of the cuboid to be painted yellow.

Based on the above information, answer the following questions.

1. Find the capacity of the hemispherical bowl.
2. Find the curved surface area of the cylindrical jar.
3. If the hemispherical bowl (filled completely with water) is to be emptied into cylindrical jars then how many cylindrical jars are required?
4. If in the cylindrical jar full of water, a conical funnel of same height and same diameter is immersed then how much water will flow out of the jar?

Based on the above information, answer the following questions.

1. Find the surface area of one such dimple.
2. Find the volume of the material dug out to make one dimple.
3. Find the total surface area exposed to the surroundings.
4. Find the volume of the golf ball.

Assertion (A): The mean of the first ten natural numbers is 5.5.

Reason (R): The sum of the first ten natural numbers is $$\frac{n(n + 1)}{2}$$.

Assertion (A): If the mean of 6, 7, p, 8, q and 14 is 9 then p + q = 19.

Reason (R): $$\text{Mean} = \dfrac{\text{sum of observations}}{\text{number of observations}}.$$

Assertion (A): The mode of the numbers 2, 3, 3, 4, 5, 4, 4, 5, 3, 4, 2, 6, 7 is 4.

Reason (R): Mode is a measure of central tendency that gives the value of the middle-most observation in the data.

Assertion (A): If the mean and median of a distribution are 169 and 170 respectively then its mode is 172.

Reason (R): The empirical relation between mean, median and mode is mode = 3 median – 2 mean.

Assertion (A): The median of the first eight prime number is 9.

Reason (R): For a data having n observations, if n is odd then the median is the $$\left(\dfrac{n + 1}{2}\right)\text{th}$$ observation.

Assertion (A): If the classes of a frequency distribution are 1–9, 10–18, 19–27, ..., etc., then the class size is 9.

Reason (R): The frequency of each class interval is centred around its midpoint.

Assertion (A): For the following frequency distribution,

the modal class is 30–40.

Reason (R): In a frequency distribution, the class having maximum frequency is called the modal class.

Assertion (A): For the following frequency distribution,

the median class is 60–80.

Reason (R): If the sum of frequencies in a continuous distribution is n then the class interval whose cumulative frequency is just greater than $$\dfrac{n}{2}$$ is called the median class.

Assertion (A): In the following frequency distribution,

the sum of the lower limit of the modal class and the upper limit of the median class is 240.

Reason (R): The median class of the given frequency distribution is 130–145.

Assertion (A): If the value of each observation of a statistical data consisting of n terms is increased by 3 then the mean of the data increases by 3n.

Reason (R): The sum of the differences between individual observations and their mean is zero.

Assertion (A): A 'more than' ogive has а descending slope while a 'less than' ogive has an ascending slope.

Reason (R): The cumulative frequencies decrease in a 'more than' type of distribution and increase in a 'less than' type of distribution.

Assertion (A): If a 'less than' type and 'more than' type cumulative frequency curves of a grouped data intersect at (9, 12) then the median of the data is 9.

Reason (R): The ordinate of the point of intersection of 'less than' type and 'more than' type cumulative frequency curves of a grouped data gives its median.

Based on the above information, answer the following questions.

1. Write the modal class.
2. Find the median of the given data.
3. Find the mean rainfall in this season.
4. If a subdivision having at least 1000 mm rainfall during monsoon season is considered as a good rainfall subdivision then how many subdivision had good rainfall?

Based on the above information, answer the following questions.

1. How many parcels have weights in the range of 300–400 grams?
2. How many parcels have weights in the range of 200–300 grams?
3. In which weight range the number of parcels is the lowest?
4. Find the mean weight of parcels received on that particular day.

Based on the above information, answer the following questions.

1. Which age group had the highest number of participants enrolled for the camp?
2. Find the modal age of people enrolled for the camp.
3. Find the median age of people enrolled for the camp.
4. If x more people of age group 55–65 had enrolled for the camp, the mean age would have been 57. Find the value of x.

Assertion (A): An event is very likely to happen if its probability is 0.99.

Reason (R): Probability of a sure event is always 1.

Assertion (A): \[\frac{7}{0.01}\] cannot be the probability of occurrence of an event.

Reason (R): The probability of an impossible event is 0.

Assertion (A): Two players, Tanya and Manya, play a table tennis match. The probability of Tanya winning the match is 0.79 and that of Manya winning the match is 0.21.

Reason (R): The sum of the probabilities of two complementary events is 1.

Assertion (A): The probability of randomly drawing a card with an even number from a box containing cards numbered 1 to 100 is \[\frac{1}{2}\].

Reason (R): \[P\text{ (event)}=\frac{\text { number of favourable outcomes }}{\text { total number of possible outcomes }}\]

Assertion (A): A bag contains 5 red, 8 black and 7 green balls. One ball is drawn at random from the bag. The probability of getting neither a black nor a red ball is \[\frac{7}{20}\].

Reason (R): The total number of balls in the bag is 20, out of which 7 are neither black nor red.

Assertion (A): If three coins are tossed simultaneously then the probability of getting at most one tail is \[\frac{1}{2}\].

Reason (R): Each of the three coins can show either of the two possibilities, i.e., head or tail.

Assertion (A): When two coins are tossed together, the probability of getting no tail is \[\frac{1}{4}\].

Reason (R): The probability P(E) of an event satisfies \[0\leq P(E)\leq 1\].

Assertion (A): A card is drawn at random from a well-shuffled pack of 52 cards. The probability that the card drawn is not an ace is \[\frac{9}{13}\].

Reason (R): When randomly drawing out a card from a well-shuffled pack of cards, the event of drawing out an ace card and the event of drawing out a non-ace card are complementary events.

Assertion (A): The probability of getting a prime number, when a die is thrown once, is \[\frac{2}{3}.\]

Reason (R): On the faces of a die, prime numbers are 2, 3 and 5.

Assertion (A): The probability of getting a doublet in a single throw of a pair of dice is \[\frac{1}{6}.\]

Reason (R): For any event \[A,\ 0\leq P(A)\leq 1.\]

Assertion (A): A fair die is thrown once. The probability of getting a prime number is \[\frac{1}{2}.\]

Reason (R): A natural number is prime if it has only two factors.

Assertion (A): Two dice are thrown together. The probability that the difference of the numbers on their upper faces equals 3 is \[\frac{1}{6}.\]

Reason (R): The total number of possible outcomes when two dice are rolled simultaneously is 36.

Assertion (A): The probability that a leap year has 53 Sundays is \[\frac{2}{7}.\]

Reason (R): The probability that a non-leap year has 53 Sundays is \[\frac{5}{7}.\]

Based on the above information, answer the following questions.

1. What is the probability that the drawn ball bears the number 8?
2. What is the probability that the drawn ball bears an even number?
3. What is the probability that the drawn ball bears a number, which is a multiple of 3?
4. What is the probability that the drawn ball is solid coloured and bears an even number?

Based on the above information, answer the following questions.

1. What is the probability that a person chosen at random had type O blood group?
2. What is the probability that a person chosen at random had type AB blood group?
3. What is the probability that a person chosen at random had neither type A nor type B blood group?
4. What is the probability that a person chosen at random had either type A or type B or type O blood group?

One school is chosen at random. Then,

1. find the probability that the school chosen at random has more than 100 computers.
2. find the probability that the school chosen at random has 50 or fewer computers.
3. find the probability that the school chosen at random has no more than 20 computers.
4. find the probability that the school chosen at random has 10 or less than 10 computers.

Based on the above information, answer the following questions.

1. List all possible outcomes of the game.
2. Find the probability of ‘making purple’.
3. For each win, a participant gets ₹ 10, but if he/she loses, he/she has to pay ₹ 5 to the school. If 99 participants played, calculate how much fund the school could have collected.
4. If the same amount of ₹ 5 has been decided for winning or losing the game then how much fund had been collected by the school? (Number of participants (= 99).

1. If a student is chosen at random then find the probability of his/her favourite colour being white.
2. What is the probability of his/her favourite colour being blue or green?
3. What is the probability of the favourite colour being red or blue?
4. If 15 students liked the colour yellow, how many students participated in the survey?